Steiner’s proof of what is now called the Steiner-Lehmus theorem was published in 1844, the same year as the book The three musketeers, written by the French author Alexandre Dumas. The motto One for all, all for one (Einer für alle, alle für einen; Un pour tous, tous pour un; Uno per tutti, tutti per uno) of the three musketeers came into widespread use in Europe in the 19th century, and its essence is that the three musketeers are inseparable; each member pledges to support the group, and the group supports each member. Now there are three classical geometries of constant curvature, namely Euclidean, spherical and hyperbolic geometries, and one can argue that, like the three musketeers, these geometries should be considered as being inseparable; that is, an idea, theorem or proof in any one of them should automatically be considered in the other two. The issue here should be not only to decide whether a particular result is true, or false, in a given geometry, but to understand which particular properties of the geometries make it so.

This Article is on the discrete Fourier transform (DFT) and the fast Fourier transform (FFT). As we shall see, FFT is a slight misnomer, causing confusion to beginners. The idiosyncratic title will be clarified in §4.

Computing machines are highly efficient nowadays, and much of the efficiency is based on the use of the FFT to speed up calculations in ultrahigh precision arithmetic. The algorithm is now an indispensable tool for solving problems that involve a large amount of computation, resulting in many useful and important applications: for example, in signal processing, data compression and photo-images in general, and WiFi, mobile phones, CT scanners and MR imaging in particular.